Emergence of Gaussianity in the thermodynamic limit of interacting fermions


Systems of interacting fermions can give rise to ground states whose correlations become effectively free-fermion-like in the thermodynamic limit, as shown by Baxter for a class of integrable models that include the one-dimensional XYZ spin-12 chain. Here, we quantitatively analyse this behaviour by establishing the relation between system size and correlation length required for the fermionic Gaussianity to emerge. Importantly, we demonstrate that this behaviour can be observed through the applicability of Wick’s theorem and thus it is experimentally accessible. To establish the relevance of our results to possible experimental realisations of XYZ-related models, we demonstrate that the emergent Gaussianity is insensitive to weak variations in the range of interactions, coupling inhomogeneities and local random potentials.

Determination of dynamical quantum phase transitions in strongly correlated many-body systems using Loschmidt cumulants

Accepted to PRX.

Dynamical phase transitions extend the notion of criticality to non-stationary settings and are characterized by sudden changes in the macroscopic properties of time-evolving quantum systems. Investigations of dynamical phase transitions combine aspects of symmetry, topology, and non-equilibrium physics, however, progress has been hindered by the notorious difficulties of predicting the time evolution of large, interacting quantum systems. Here, we tackle this outstanding problem by determining the critical times of interacting many-body systems after a quench using Loschmidt cumulants. Specifically, we investigate dynamical topological phase transitions in the interacting Kitaev chain and in the spin-1 Heisenberg chain. To this end, we map out the thermodynamic lines of complex times, where the Loschmidt amplitude vanishes, and identify the intersections with the imaginary axis, which yield the real critical times after a quench. For the Kitaev chain, we can accurately predict how the critical behavior is affected by strong interactions, which gradually shift the time at which a dynamical phase transition occurs. Our work demonstrates that Loschmidt cumulants are a powerful tool to unravel the far-from-equilibrium dynamics of strongly correlated many-body systems, and our approach can immediately be applied in higher dimensions.


Lee-Yang theory, high cumulants, and large-deviation statistics of the magnetization in the Ising model

Editors’ Suggestion: Phys. Rev. B 102, 174418 (2020)

We investigate the Ising model in one, two, and three dimensions using a cumulant method that allows us to determine the Lee-Yang zeros from the magnetization fluctuations in small lattices. By doing so with increasing system size, we are able to determine the convergence point of the Lee-Yang zeros in the thermodynamic limit and thereby predict the occurrence of a phase transition. The cumulant method is attractive from an experimental point of view since it uses fluctuations of measurable quantities, such as the magnetization in a spin lattice, and it can be applied to a variety of equilibrium and nonequilibrium problems. We show that the Lee-Yang zeros encode important information about the rare fluctuations of the magnetization. Specifically, by using a simple ansatz for the free energy, we express the large-deviation function of the magnetization in terms of Lee-Yang zeros. This result may hold for many systems that exhibit a first-order phase transition.


Show BibTeX: @article{Deger2020b,
title = {{L}ee-{Y}ang theory, high cumulants, and large-deviation statistics of the magnetization in the {I}sing model},
author = {Deger, Aydin and Brange, Fredrik and Flindt, Christian},
journal = {Phys. Rev. B},
volume = {102},
issue = {17},
pages = {174418},
numpages = {12},
year = {2020},
month = {Nov},
publisher = {American Physical Society},
doi = {10.1103/PhysRevB.102.174418},
url = {https://link.aps.org/doi/10.1103/PhysRevB.102.174418}}

Lee-Yang theory of the Curie-Weiss model and its rare fluctuations

Open access: Phys. Rev. Research 2, 033009 (2020)

Phase transitions are typically accompanied by nonanalytic behaviors of the free energy, which can be explained by considering the zeros of the partition function in the complex plane of the control parameter and their approach to the critical value on the real axis as the system size is increased. Recent experiments have shown that partition function zeros are not just a theoretical concept. They can also be determined experimentally by measuring fluctuations of thermodynamic observables in systems of finite size. Motivated by this progress, we investigate here the partition function zeros for the Curie-Weiss model of spontaneous magnetization using our recently established cumulant method. Specifically, we extract the leading Fisher and Lee-Yang zeros of the Curie-Weiss model from the fluctuations of the energy and the magnetization in systems of finite size. We develop a finite-size scaling analysis of the partition function zeros, which is valid for mean-field models and which allows us to extract both the critical values of the control parameters and the critical exponents, even for small systems that are away from criticality. We also show that the Lee-Yang zeros carry important information about the rare magnetic fluctuations as they allow us to predict many essential features of the large-deviation statistics of the magnetization. This finding may constitute a profound connection between Lee-Yang theory and large-deviation statistics.


Show BibTeX: @article{Deger2020,
title = {{L}ee-{Y}ang theory of the {C}urie-{W}eiss model and its rare fluctuations},
author = {Deger, Aydin and Flindt, Christian},
journal = {Phys. Rev. Research},
volume = {2},
issue = {3},
pages = {033009},
numpages = {10},
year = {2020},
month = {Jul},
publisher = {American Physical Society},
doi = {10.1103/PhysRevResearch.2.033009},
url = {https://link.aps.org/doi/10.1103/PhysRevResearch.2.033009}}

Determination of Universal Critical Exponents Using Lee-Yang Theory

Open access: Physical Review Research 1, 023004 (2019)

Lee-Yang zeros are points in the complex plane of an external control parameter at which the partition function vanishes for a many-body system of finite size. In the thermodynamic limit, the Lee-Yang zeros approach the critical value on the real axis, where a phase transition occurs. Partition function zeros have for many years been considered a purely theoretical concept; however, the situation is changing now as Lee-Yang zeros have been determined in several recent experiments. Motivated by these developments, we here devise a direct pathway from measurements of partition function zeros to the determination of critical points and universal critical exponents of continuous phase transitions. To illustrate the feasibility of our approach, we extract the critical exponents of the Ising model in two and three dimensions from the fluctuations of the total energy and the magnetization in lattices of finite size. Importantly, the critical exponents can be determined even if the system is away from the phase transition. Moreover, in contrast to standard methods based on Binder cumulants, it is not necessary to drive the system across the phase transition. As such, our method provides an intriguing perspective for investigations of phase transitions that may be hard to reach experimentally, for instance at very low temperatures or at very high pressures.


Show BibTeX: @article{Deger2019,
title = {Determination of universal critical exponents using {L}ee-{Y}ang theory},
author = {Deger, Aydin and Flindt, Christian},
journal = {Phys. Rev. Research},
volume = {1},
issue = {2},
pages = {023004},
numpages = {7},
year = {2019},
month = {Sep},
publisher = {American Physical Society},
doi = {10.1103/PhysRevResearch.1.023004},
url = {https://link.aps.org/doi/10.1103/PhysRevResearch.1.023004}}